# Complex variables conformal mapping trig identity

1. Apr 19, 2010

### EnginerdRuns

1. The problem statement, all variables and given/known data
map the function $$$$w = \Big(\frac{z-1}{z+1}\Big)^{2}$$$$
on some domain which contains $$z=e^{i\theta}$$. $$\theta$$ between 0 and $$\pi$$
Hint: Map the semicircular arc bounding the top of the disc by putting $$z=e^{i\theta}$$ in the above formula. The resulting expression reduces to a simple trig function.

2. Relevant equations
I can get the map if I can figure out what function they're going for, but I have no idea what function this is.

3. The attempt at a solution
$$w = \Big(\frac{e^{i\theta}-1}{e^{i\theta}+1}\Big)^{2}$$
Where the heck do I go from here?

2. Apr 19, 2010

### tiny-tim

Welcome to PF!

Hi EnginerdRuns! Welcome to PF!

Multiply top and bottom of the fraction by e-iθ/2

3. Apr 19, 2010

### EnginerdRuns

Thanks bro. I'm running off of way too little sleep at this point.